TRILINEAR COORDINATES

Suppose P is a point in the plane of a triangle ABC. Any three numbers proportional to the directed distances from P to the sidelines BC, CA, AB, in that order, are called homogeneous trilinear coordinates for P. Often the word homogeneous is omitted, or the whole thing is shortened to trilinears.

The incenter of triangle ABC, for example, being equidistant from the sidelines, has trilinears 1 : 1 : 1. The vertices themselves are given especially simply by

A = 1 : 0 : 0,
B = 0 : 1 : 0,
C = 0 : 0 : 1.

The best known triangle centers all have remarkably simple trilinears. See for yourself: